Differential Geometry Exercises I: Tensors - Fau.edu

5. [Schutz 3.5, p. 81]
a. Show that if ¯V and ¯W are linear combinations (not necessarily with constant coefficients)
of m vector fields that all commute with one another, then the Lie bracket of ¯V and ¯W
is a linear combination of the same m fields.
b. Prove the same result when the m vector fields have Lie brackets which are nonvanishing
linear combinations of the m fields.
6. Let ξ a be a Killing vector field for the metric g ab , and let η a be the tangent vector to a geodesic of
g ab in an affine parameterization. Show that the inner product of these vectors is constant along
the geodesic. What happens to this conserved quantity if one changes affine parameterizations?
What happens if the parameterization is not affine?
7. Let g ab be a stationary spacetime metric — meaning that it has a time-like Killing field t a —
that solves the vacuum Einstein equations R ab = 0.
a. Show that F ab := ∇ a t b satisfies the source-free Maxwell equations.
b. Suppose that t a is in fact a static Killing field — meaning that it is orthogonal everywhere
to some space-like surfaces Σ. Calculate the electric and magnetic parts of F ab
on the static slices Σ.
c. Use the Gauss law to compute the “electric charge” of the Schwarzschild metric
ds 2 = (1 − 2M/r) dt 2 + (1 − 2M/r) −1 dr 2 + r 2 (dθ 2 + sin 2 θ dφ 2 ).
Note that the static Killing field in this case is ∂ t , and that the static slices are the
surfaces of constant t in spacetime.
Hint: The electric charge is given by a flux integral. Show that one gets the same result
no matter which two-sphere one uses in the integral. Then, calculate in the asymptotic
region where r → ∞.